Some extensions for Ramanujan's circular summation formula

TL;DR

This paper extends Ramanujan's circular summation formula to include mixed products of two Jacobi theta functions and derives new identities, broadening the understanding of theta function relationships.

Contribution

It introduces novel extensions of Ramanujan's formula involving mixed theta function products and presents new identities of Jacobi's theta functions.

Findings

Extended Ramanujan's circular summation formula with mixed products

Derived new identities of Jacobi's theta functions

Enhanced understanding of theta function relationships

Abstract

In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.

Equations

$${\vartheta }_1\left(z|\tau \right)$$

$${\vartheta }_2\left(z|\tau \right)$$

$${\vartheta }_3\left(z|\tau \right)$$

$${\vartheta }_4\left(z|\tau \right)$$

$${\vartheta }_1\left(z+\pi |\tau \right)=-{\vartheta }_1\left(z|\tau \right),{\vartheta }_1\left(z+\pi \tau |\tau \right)=-q^{-1}{e}^{-2iz}{\vartheta }_1\left(z|\tau \right),$$

$${\vartheta }_2\left(z+\pi |\tau \right)=-{\vartheta }_2\left(z|\tau \right),{\vartheta }_2\left(z+\pi \tau |\tau \right)=q^{-1}{e}^{-2iz}{\vartheta }_2\left(z|\tau \right),$$

$${\vartheta }_3\left(z+\pi |\tau \right)={\vartheta }_3\left(z|\tau \right),{\vartheta }_3\left(z+\pi \tau |\tau \right)=q^{-1}{e}^{-2iz}{\vartheta }_3\left(z|\tau \right),$$

$${\vartheta }_4\left(z+\pi |\tau \right)={\vartheta }_4\left(z|\tau \right),{\vartheta }_4\left(z+\pi \tau |\tau \right)=-q^{-1}{e}^{-2iz}{\vartheta }_4\left(z|\tau \right).$$

$${\vartheta }_1\left(z+n\pi \tau |\tau \right)=(-1{)}^n{q}^{-n^2}{e}^{-2niz}{\vartheta }_1\left(z|\tau \right),$$

$${\vartheta }_2\left(z+n\pi \tau |\tau \right)=q^{-n^2}{e}^{-2niz}{\vartheta }_2\left(z|\tau \right),$$

$${\vartheta }_3\left(z+n\pi \tau |\tau \right)=q^{-n^2}{e}^{-2niz}{\vartheta }_3\left(z|\tau \right),$$

$${\vartheta }_4\left(z+n\pi \tau |\tau \right)=(-1{)}^n{q}^{-n^2}{e}^{-2niz}{\vartheta }_4\left(z|\tau \right).$$

$$\sum _{-n/2<r\le n/2}{\left(\sum _{\begin{array}{c}k=-\infty \\ k\equiv r(\text{mod}n)\end{array}}^{\infty }{a}^{k(k+1)/(2n)}{b}^{k(k-1)/(2n)}\right)}^n=f(a,b)F_n(ab),$$

$$F_n(q):=1+2n{q}^{(n-1)/2}+\cdots ,n\ge 3.$$

$$f(a,b)=\sum _{n=-\infty }^{\infty }{a}^{n(n+1)/2}{b}^{n(n-1)/2},\left|ab\right|<1.$$

$$\sum _{k=0}^{n-1}{U}_k^n{f}^n\left(\frac{U_{n+k}}{U_k},\frac{V_{n-k}}{V_k}\right)=f(a,b)F_n(ab),$$

$$F_n(q):=1+2n{q}^{(n-1)/2}+\cdots ,n\ge 3.$$

$$\sum _{k=0}^{n-1}{q}^{k^2}{e}^{2kiz}{\vartheta }_3^n(z+k\pi \tau |n\tau )={\vartheta }_3(z|\tau )F_n(\tau ).$$

$$F_n(q)=1+2n{q}^{n-1}+\cdots .$$

$$\sum _{k=0}^{n-1}{\vartheta }_3^n\left(z+\frac{k\pi }{n}|\tau \right)=G_n(\tau ){\vartheta }_3(nz|n\tau ),$$

$$G_n(\tau )=n\sum _{\begin{array}{c}{r}_1+\cdots +r_n=0\\ r_1,\dots ,r_n=-\infty \end{array}}^{\infty }{q}^{r_1^2+\cdots +r_n^2}=\sqrt{n}(-i\tau {)}^{(1-n)/2}{F}_n\left(-\frac{1}{n\tau }\right).$$

$$\sum _{k=0}^{n-1}{\vartheta }_3(z+k\pi |\tau )=n{\vartheta }_3(nz|n^2\tau ).$$

$$\sum _{s=0}^{kn-1}{\vartheta }_3^a\left(\frac{z}{kn}+\frac{y}{a}+\frac{\pi s}{kn}|\frac{\tau }{k{n}^2}\right){\vartheta }_3^b\left(\frac{z}{kn}-\frac{y}{b}+\frac{\pi s}{kn}|\frac{\tau }{k{n}^2}\right)=R_{33}\left(a,b;\frac{y}{ab},\frac{\tau }{k{n}^2}\right){\vartheta }_3\left(z|\tau \right),$$

$$R_{33}\left(a,b;y,\tau \right)=kn\sum _{\begin{array}{c}{m}_1+\cdots +m_a+n_1+\cdots +n_b=0\\ m_1,\cdots ,m_a,n_1,\cdots ,n_b=-\infty \end{array}}^{\infty }{q}^{m_1^2+\cdots +m_a^2+n_1^2+\cdots +n_b^2}{e}^{2k(m_1+\cdots +m_a)iy}.$$

$$\sum _{k=0}^{mn-1}\prod _{j=1}^n{\vartheta }_3\left(z+y_j+\frac{k\pi }{mn}|\tau \right)=G_{m,n}(y_1,y_2,\cdots ,y_n|\tau ){\vartheta }_3(mnz|m^{2}n\tau ),$$

$$G_{m,n}(y_1,y_2,\cdots ,y_n|\tau )=mn\sum _{\begin{array}{c}{r}_1,\cdots ,r_n=-\infty \\ r_1+\cdots +r_n=0\end{array}}^{\infty }{q}^{r_1^2+r_2^2+\cdots +r_n^2}{e}^{2i(r_1{y}_1+r_2{y}_2+\cdots +r_n{y}_n)}.$$

$$\sum _{k=0}^{mn-1}\prod _{j=1}^a{\vartheta }_1\left(z+x_j+\frac{k\pi }{mn}|\tau \right)\prod _{i=1}^b{\vartheta }_2\left(z+y_i+\frac{k\pi }{mn}|\tau \right)=R_{1,2}\left(m,n;\tau \right){\vartheta }_3\left(mnz|m^{2}n\tau \right).$$

$$\sum _{k=0}^{mn-1}\prod _{j=1}^a{\vartheta }_1\left(z+x_j+\frac{k\pi }{mn}|\tau \right)\prod _{i=1}^b{\vartheta }_2\left(z+y_i+\frac{k\pi }{mn}|\tau \right)=R_{1,2}\left(m,n;\tau \right){\vartheta }_4\left(mnz|m^{2}n\tau \right),$$

$$R_{1,2}\left(m,n;\tau \right)=$$

$$\times q^{r_1^2+\cdots +r_a^2+s_1^2+\cdots +s_b^2}{e}^{2i\left(r_1{x}_1+\cdots +r_a{x}_a+s_1{y}_1+\cdots +s_b{y}_b\right)}.$$

$$\sum _{k=0}^{mn-1}\prod _{j=1}^a{\vartheta }_1\left(z+x_j+\frac{k\pi }{mn}|\tau \right)\prod _{i=1}^b{\vartheta }_3\left(z+y_i+\frac{k\pi }{mn}|\tau \right)=R_{1,3}\left(m,n;\tau \right){\vartheta }_3\left(mnz|m^{2}n\tau \right),$$

$$R_{1,3}\left(m,n;\tau \right)=$$

$$\times q^{r_1^2+\cdots +r_a^2+s_1^2+\cdots +s_b^2+r_1+\cdots +r_a}{e}^{i\left(2r_1{x}_1+\cdots +2r_a{x}_a+2s_1{y}_1+\cdots +2s_b{y}_b+x_1+\cdots +x_a\right)}.$$

$$\sum _{k=0}^{mn-1}\prod _{j=1}^a{\vartheta }_1\left(z+x_j+\frac{k\pi }{mn}|\tau \right)\prod _{i=1}^b{\vartheta }_4\left(z+y_i+\frac{k\pi }{mn}|\tau \right)=R_{1,4}\left(m,n;\tau \right){\vartheta }_3\left(mnz|m^{2}n\tau \right).$$

$$\sum _{k=0}^{mn-1}\prod _{j=1}^a{\vartheta }_1\left(z+x_j+\frac{k\pi }{mn}|\tau \right)\prod _{i=1}^b{\vartheta }_4\left(z+y_i+\frac{k\pi }{mn}|\tau \right)=R_{1,4}\left(m,n;\tau \right){\vartheta }_4\left(mnz|m^{2}n\tau \right),$$

$$R_{1,4}\left(m,n;\tau \right)=$$

Full text

Some extensions for Ramanujan’s circular summation formula

**Ji-Ke Ge1 and Qiu-Ming Luo2,∗

** 1School of Electrical and Information Engineering

Chongqing University of Science and Technology

Chongqing Higher Education Mega Center, Huxi Campus

Chongqing 401331, People’s Republic of China

**E-Mail: gjkweb@126.com

** 2Department of Mathematics, Chongqing Normal University

Chongqing Higher Education Mega Center, Huxi Campus

Chongqing 401331, People’s Republic of China

**E-Mail: luomath2007@163.com

** *∗*Corresponding author

Abstract

In this paper, we give some extensions for Ramanujan’s circular summation formula with the mixed products of two Jacobi’s theta functions. As some applications, we also obtain many interesting identities of Jacobi’s theta functions.

**2010 Mathematics Subject Classification. ** Primary 11F27; Secondary 11F20, 33E05.

**Key Words and Phrases. ** Elliptic functions; Jacobi’s theta functions; Ramanujan’s circular summation; identities of Jacobi’s theta functions.

1. Introduction and main results

Throughout this paper we take $q=e^{\pi{i}\tau}$, $\Im(\tau)>0$, $z\in\mathbb{C}$.

The classical Jacobi’s theta functions $\vartheta_{i}\left(z|\tau\right),i=1,2,3,4,$ are defined by

[TABLE]

We have

[TABLE]

By applying the induction, we easily obtain

[TABLE]

On page 54 in Ramanujan’s lost notebook (see [22, p. 54, Entry 9.1.1 ], or [2, p. 337]), Ramanujan recorded the following claim (without proof) which is now well known as Ramanujan’s circular summation. The appellation circular summation was initiated by Son (see [2, p. 338]).

Theorem 1.1** (Ramanujan’s circular summation).**

For each positive integer $n$ and $\left|ab\right|<1$,

[TABLE]

where

[TABLE]

Ramanujan’s theta function $f(a,b)$ is defined by

[TABLE]

By the definition of Ramanujan’s theta function above and routine calculations, we can rewrite Ramanujan’s circular summation (1.13) as follows (see, for details [2, p. 338]).

Theorem 1.2** (Ramanujan’s circular summation).**

Let $U_{k}=a^{{k(k+1)}/{(2n)}}$ and $V_{k}=b^{{k(k-1)}/{(2n)}}$. For each positive integer $n$ and $\left|ab\right|<1$, then

[TABLE]

where

[TABLE]

When $n=1$, the identity (1.13) of Theorem 1.1 merely reduces to the tautology $f(a,b)=f(a,b)$. When $n=2$, the identity (1.13) of Theorem 1.1 holds if the coefficient 2 in (1.14) is deleted.

The claim has been first proven by Rangachari [23] who also verified Ramanujan’s explicit and elegant formulae for $F_{n}(x)$ for $n=2,3,4,5,7$ by employing Mumford’s theory of theta functions and root lattices. Several authors have determined the identification of $F_{n}(x)$ in further special cases. S. Ahlgren [1] considered the cases $n=6,8,9$, and K. Ono [21] established $F_{11}(x)$, while K.S. Chua [15] derived the corresponding result for $F_{13}(x)$. K.S. Chua [14] and T. Murayama [20], independently, improved the work of Rangachari by removing a condition of primality from Rangachari’s work. S.H. Son [25] devised a proof of (1.13) that is more in tune with Ramanujan’s work, Son used functional equations in the spirit of $q$-series. A summary of all known dentifications of $F_{n}(x)$ can be found in Son’s paper [25].

If we are going to apply the transformation $\tau\longmapsto-\frac{1}{\tau}$ to Ramanujan’s identity, it will be convenient to convert Ramanujan’s theorem into one involving the classical theta function $\vartheta_{3}(z|\tau)$ defined by (1.3). H. H. Chan, Z.-G. Liu and S. T. Ng [12] prove that Theorem 1.3 below is equivalent to Theorem 1.1.

Theorem 1.3** (Ramanujan’s circular summation).**

For any positive integer $n\geq 2$, we have

[TABLE]

When $n\geq 3$,

[TABLE]

H. H. Chan, Z.-G. Liu and S. T. Ng [12] also showed that Theorem 1.4 below is equivalent to Theorem 1.1, Theorem 1.2 and Theorem 1.3 by applying Jacobi’s imaginary transformation formula [26, p. 475].

Theorem 1.4** (Ramanujan’s circular summation).**

For any positive integer $n$, we have

[TABLE]

where

[TABLE]

Many years ago, M. Boon et al. [6, p. 3440, Eq. (7)] obtained the following additive decomposition of $\vartheta_{3}(z|\tau)$.

Theorem 1.5**.**

For any positive integer $n$, we have

[TABLE]

Recentely, Zeng [28] proved the following theorem which unifies Theorem 1.4 and Theorem 1.5.

Theorem 1.6**.**

For any nonegative integer $k,n,a$ and $b$ with $a+b=n$,

[TABLE]

where

[TABLE]

S. H. Chan and Z.-G. Liu [11] further extended Zeng’s result as the following form.

Theorem 1.7**.**

Suppose $y_{1},y_{2},\cdots,y_{n}$ are $n$ complex numbers such that $y_{1}+y_{2}+\cdots+y_{n}=0$,

[TABLE]

where

[TABLE]

Clearly, if $a$ and $b$ are two positive integers such that $a+b=n$, then we can take $y_{1}=y_{2}=\cdots=y_{a}=y/a$ and $y_{a+1}=y_{a+2}=\cdots=y_{n}=-y/b$ in Theorem 1.7 to obtain Theorem 1.6.

Zhu [29, 30] study the alternating circular summation formula of theta functions and also correct an error of [11]. Cai et al. [8, 9] obtain other circular summation formulas of theta functions. Z.-G. Liu [18, p. 1978, Theorem 1.1 and Theorem 1.2] further obtained two theorems which unify Theorem 1.7 and [29, p. 117, Theorem 1.7] of Zhu.

In the present paper, motivated by [8], [11] and [12], we further give some extensions for Ramanujan’s circular summation by applying the method of elliptic function. We also give some applications of the circular summation formulas. We now state our results as follows.

Theorem 1.8**.**

Let $n$ be even, $m$ be any positive integers, $a$ and $b$ be any non-negative integers such that $a+b=n$, and $x_{1},\ldots,x_{a},y_{1},\ldots,y_{b}$ be any complex numbers such that their sum be [math].

• •

When $ma$ is even, we have

[TABLE]

• •

When $ma$ is odd, we have

[TABLE]

where

[TABLE]

Theorem 1.9**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be any non-negative integers such that $a+b=n$, and $x_{1},\ldots,x_{a},y_{1},\ldots,y_{b}$ be any complex numbers such that their sum be [math]. We have

[TABLE]

where

[TABLE]

Theorem 1.10**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be any non-negative integers such that $a+b=n$, and $x_{1},\ldots,x_{a},y_{1},\ldots,y_{b}$ be any complex numbers such that their sum be [math].

• •

When $mn$ is even, we have

[TABLE]

• •

When $mn$ is odd, we have

[TABLE]

where

[TABLE]

Theorem 1.11**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be any non-negative integers such that $a+b=n$, and $x_{1},\ldots,x_{a},y_{1},\ldots,y_{b}$ be any complex numbers such that their sum be [math]. We have

[TABLE]

where

[TABLE]

Theorem 1.12**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be any non-negative integers such that $a+b=n$, and $x_{1},\ldots,x_{a},y_{1},\ldots,y_{b}$ be any complex numbers such that their sum be [math].

• •

When $mb$ is even, we have

[TABLE]

• •

When $mb$ is odd, we have

[TABLE]

where

[TABLE]

Theorem 1.13**.**

Let $m$ and $n$ be any positive integers, $a$ and $b$ be any non-negative integers such that $a+b=n$, and $x_{1},\ldots,x_{a},y_{1},\ldots,y_{b}$ be any complex numbers such that their sum be [math].

• •

When $mb$ is even, we have

[TABLE]

• •

When $mb$ is odd, we have

[TABLE]

where

[TABLE]

2. Proofs of the main results

In this section, we only prove Theorem 1.8, the proofs of other theorems are similar.

Let $f(z)$ be the left-hand of (1.26) with $z\longmapsto\frac{z}{mn},\tau\longmapsto\frac{\tau}{m^{2}n}$, we have

[TABLE]

By (1.5) and (1.6) and noting that $a+b=n$, we easily obtain

[TABLE]

When $n$ is even, comparing (2.1) and (2), we get

[TABLE]

By (1.9) and (1.10), noting that $x_{1}+\cdots+x_{a}+y_{1}+\cdots+y_{b}=0$ and $a+b=n$, we have

[TABLE]

$\bullet$ When $ma$ is even in (2), we have

[TABLE]

We construct the function $\dfrac{f(z)}{\vartheta_{3}(z|\tau)}$, by (1.7), (2.3) and (2.5), we find that the function $\dfrac{f(z)}{\vartheta_{3}(z|\tau)}$ is an elliptic function with the double periods $\pi$ and $\pi\tau$, and only have a simple pole at $z=\dfrac{\pi}{2}+\dfrac{\pi\tau}{2}$ in the period parallelogram. Hence the function $\dfrac{f(z)}{\vartheta_{3}(z|\tau)}$ is a constant, say this constant is ${C}_{1,2}^{(1)}(m,n;\tau)$, i.e.,

[TABLE]

we have

[TABLE]

By (2.1) and (2.6), we obtain

[TABLE]

Letting

[TABLE]

in (2.7), and then setting

[TABLE]

we arrive at (1.26).

Below we calculate the constant $R_{1,2}^{(1)}\left(m,n;\tau\right)$.

Setting

[TABLE]

in (1.1), by some simple calculation and noting that $n$ is even, we obtain

[TABLE]

Setting

[TABLE]

in (1.2), we obtain

[TABLE]

Setting

[TABLE]

in (1.3), we get

[TABLE]

Substituting (2.8), (2.9) and (2.10) into (1.26), noting that $x_{1}+\cdots+x_{a}+y_{1}+\cdots+y_{b}=0$ and $a+b=n$, we find

[TABLE]

Equating the constant of both sides of (2), noting that $r_{1}+\cdots+r_{a}+s_{1}+\cdots+s_{b}=-\frac{n}{2}$, we have

[TABLE]

$\bullet$ When $ma$ is odd in (2), we have

[TABLE]

We construct the function $\frac{f(z)}{\vartheta_{4}(z|\tau)}$, by (1.8), (2.5) and (2.13), we find that the function $\frac{f(z)}{\vartheta_{4}(z|\tau)}$ is an elliptic function with double periods $\pi$ and $\pi\tau$, and has only a simple pole at $z=\frac{\pi\tau}{2}$ in the period parallelogram. Hence the function $\frac{f(z)}{\vartheta_{4}(z|\tau)}$ is a constant, say $C_{1,2}^{(2)}(m,n;\tau)$, we have

[TABLE]

or, equivalently

[TABLE]

Letting

[TABLE]

in (2.14), and then setting

[TABLE]

we arrive at (1.27).

Similarly, by (1.1), (1.2) and (1.4), from (1.27), we obtain

[TABLE]

Equating the constant of both sides of (2), we have

[TABLE]

Clearly, we have

[TABLE]

The proof is complete.

3. Circular summation formulas for the products of the single Jacobi’s theta functions

In this section, we deduce the circular summation formulas for the products of the single Jacobi’s theta functions from the above section.

Theorem 3.1**.**

Suppose that $n$ is even, $m$ is any positive integers; $y_{1},y_{2},\ldots,y_{n}$ are any complex numbers such that $y_{1}+y_{2}+\cdots+y_{n}=0$, we have

[TABLE]

where

[TABLE]

Proof.

Define that the empty product \prod_{j=1}^{b}=1,\textup{when b<1}. Setting $a=n$ and $b=0$ in (1.26) of Theorem 1.8, noting that $r_{1}+\cdots+r_{n}=\frac{n}{2}$ and $n$ is even, we deduce Theorem 3.1. ∎

Theorem 3.2**.**

Suppose that $n$ is even, $m$ is any positive integers; $y_{1},y_{2},\ldots,y_{n}$ are any complex numbers such that $y_{1}+y_{2}+\cdots+y_{n}=0$, we have

[TABLE]

where

[TABLE]

Proof.

Define that the empty product \prod_{j=1}^{a}=1,\textup{when a<1}. Setting $a=0$ and $b=n$ in (1.26) of Theorem 1.8, noting that $s_{1}+\cdots+s_{n}=\frac{n}{2}$ and $n$ is even, we deduce Theorem 3.2. ∎

Theorem 3.3**.**

Suppose that $m,n$ are any positive integers; $y_{1},y_{2},\ldots,y_{n}$ are any complex numbers such that $y_{1}+y_{2}+\cdots+y_{n}=0$, we have

[TABLE]

where

[TABLE]

Proof.

Define that the empty product \prod_{j=1}^{b}=1,\textup{when b<1}. Setting $a=n$ and $b=0$ in (1.39) of Theorem 1.13, we deduce Theorem 3.3. ∎

Theorem 3.4**.**

Suppose that $mn$ is even; $y_{1},y_{2},\ldots,y_{n}$ are any complex numbers such that $y_{1}+y_{2}+\cdots+y_{n}=0$, we have

[TABLE]

where

[TABLE]

Proof.

Define that the empty product \prod_{j=1}^{a}=1,\textup{when a<1}. Setting $a=0$ and $b=n$ in (1.39) of Theorem 1.13, we deduce Theorem 3.4. ∎

Remark 3.5*.*

Theorem 3.3 is just the main result of Chan and Liu (see [11, p. 1191, Theorem 4]). Theorem 3.1, Theorem 3.2 and Theorem 3.4 are some analogues of the main result (Theorem 1.7) of Chan and Liu.

If taking $m=1$ and $y_{1}=y_{2}=\cdots=y_{n}=0$ in Theorem 3.1, Theorem 3.2, Theorem 3.3 and Theorem 3.4, we further deduce the following interesting results.

Corollary 3.6**.**

For $n$ is even, we have

[TABLE]

where

[TABLE]

Corollary 3.7**.**

For $n$ is even, we have

[TABLE]

where

[TABLE]

Corollary 3.8**.**

For $n$ is any positive integers, we have

[TABLE]

where

[TABLE]

Corollary 3.9**.**

For $n$ is even, we have

[TABLE]

where

[TABLE]

Remark 3.10*.*

Theorem 3.8 is just Ramanujan’s circular summation Theorem 1.20. Theorem 3.6, Theorem 3.7 and Theorem 3.9 are some analogues of Ramanujan’s circular summation.

If setting $x_{1}=x_{2}=\cdots=x_{a}=x$ and $y_{1}=y_{2}=\cdots=y_{b}=y$ in Theorem 1.8–Theorem 1.13 respectively, then we obtain the following interesting special cases.

Theorem 3.11**.**

Let $n$ be even, $m$ be any positive integers, $a$ and $b$ be non-negative integers such that $a+b=n$, and $x$ and $y$ be any complex numbers such that $ax+by=0$.

• •

When $ma$ is even, we have

[TABLE]

• •

When $ma$ is odd, we have

[TABLE]

where

[TABLE]

Theorem 3.12**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be non-negative integers such that $a+b=n$, and $x$ and $y$ be any complex numbers such that $ax+by=0$. We have

[TABLE]

where

[TABLE]

Theorem 3.13**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be non-negative integers such that $a+b=n$, and $x$ and $y$ be any complex numbers such that $ax+by=0$.

• •

When $mn$ is even, we have

[TABLE]

• •

When $mn$ is odd, we have

[TABLE]

where

[TABLE]

Theorem 3.14**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be non-negative integers such that $a+b=n$, and $x$ and $y$ be any complex numbers such that $ax+by=0$. We have

[TABLE]

where

[TABLE]

Theorem 3.15**.**

Let $a$ be even, $m$ and $n$ be any positive integers, $a$ and $b$ be non-negative integers such that $a+b=n$, and $x$ and $y$ be any complex numbers such that $ax+by=0$.

• •

When $mb$ is even, we have

[TABLE]

• •

When $mb$ is odd, we have

[TABLE]

where

[TABLE]

Theorem 3.16**.**

Let $m$ and $n$ be any positive integers, $a$ and $b$ be non-negative integers such that $a+b=n$, and $x$ and $y$ be any complex numbers such that $ax+by=0$.

• •

When $mb$ is even, we have

[TABLE]

• •

When $mb$ is odd, we have

[TABLE]

where

[TABLE]

Remark 3.17*.*

Theorem 3.11–Theorem 3.16 are some analogues of Theorem 1.6 of Zeng.

4. Some identities of Jacobi’s theta functions

In this section, we obtain some interesting identities of Jacobi’s theta functions from Theorem 1.8-Theorem 1.13.

The multiple theta series $a(y_{1},y_{2}|\tau),b(y_{1},y_{2}|\tau),c(y_{1},y_{2}|\tau),d(y_{1},y_{2}|\tau)$ are respectively defined by

[TABLE]

The corresponding cubic theta functions $a(\tau),b(\tau)$ and $c(\tau)$ are respectively defined by (see [7])

[TABLE]

• •

Applications of Theorem 1.8.

Taking $m=n=2,a=b=1$ in (1.26) of Theorem 1.8, then $x_{1}+y_{1}=0$. Setting $x=x_{1}$, by (1.8), via some direct computations, we have

[TABLE]

Hence we obtain the folllowing identity for products of two theta functions:

[TABLE]

Taking $n=2,m=a=b=1$ in (1.27) of (1.8), then $x_{1}+y_{1}=0$. Setting $x=x_{1}$, by (1.8), via some direct computations, we have

[TABLE]

Hence we obtain the folllowing identity for products of two theta functions:

[TABLE]

• •

Applications of Theorem 1.9.

Taking $n=3,a=2,b=1$ in (1.29) of Theorem 1.9, then $x_{1}+x_{2}+y_{1}=0$. By (1.9), via some direct computations, we have

[TABLE]

Hence we obtain the folllowing identity for products of three theta functions:

[TABLE]

Further taking $m=1$ and $m=2$ in (4.12) respectively, we get

[TABLE]

and

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.13) and (4.14) respectively, we get

[TABLE]

and

[TABLE]

The above $d(\tau)$ and $d(y_{1},y_{2}|\tau)$ are defined by (4.4) and (4.8) respectively.

• •

Applications of Theorem 1.10.

Taking $n=3,a=2,b=1$ in (1.31) of Theorem 1.10, then $x_{1}+x_{2}+y_{1}=0$. By (1.10), via some direct computations, we have

[TABLE]

Hence we obtain the folllowing identity for products of three theta functions:

[TABLE]

Taking $m=2$ in the first equation of (4.17), we have

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.18), we have

[TABLE]

Taking $m=1$ in the second equation of (4.17), we have

[TABLE]

The above $c(\tau)$ and $c(y_{1},y_{2}|\tau)$ are defined by (4.3) and (4.7) respectively.

• •

Applications of Theorem 1.11.

Taking $n=3,a=2,b=1$ in (1.34) of Theorem 1.11, then $x_{1}+x_{2}+y_{1}=0$. By (1.11), via some direct computations, we have

[TABLE]

Hence we obtain the folllowing identity for products of three theta functions:

[TABLE]

Taking $m=1$ and $m=2$ in (4.21) respectively, we get

[TABLE]

and

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.22) and (4.23) respectively, we get

[TABLE]

and

[TABLE]

The above $c(\tau)$ and $c(y_{1},y_{2}|\tau)$ are defined by (4.4) and (4.8) respectively.

• •

Applications of Theorem 1.12.

Taking $n=3,a=2,b=1$ in Theorem 1.12, then $x_{1}+x_{2}+y_{1}=0$. By (1.12), via some direct computations, we have

[TABLE]

Hence we obtain the folllowing identity for products of three theta functions:

[TABLE]

Taking $m=2$ in the first equation of (4.26), we get

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.18), we get

[TABLE]

Taking $m=1$ in the second equation of (4.26), we get

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.29), we get

[TABLE]

The above $d(\tau)$ and $d(y_{1},y_{2}|\tau)$ are defined by (4.4) and (4.8) respectively.

• •

Applications of Theorem 1.13.

Cases 1.   Taking $n=3,a=2,b=1$ in Theorem 1.13, then $x_{1}+x_{2}+y_{1}=0$. By (1.13), via some direct computations, we have

[TABLE]

Hence we obtain the folllowing identity for products of three theta functions:

[TABLE]

Taking $m=2$ in the first equation of (4.31), we get

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.32), we get

[TABLE]

Taking $m=1$ in the second equation of (4.32), we get

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.34), we get

[TABLE]

The above $b(\tau)$ and $b(y_{1},y_{2}|\tau)$ are defined by (4.2) and (4.6) respectively.

Cases 2.   Taking $n=3,a=1,b=2$ in Theorem 1.13, then $x_{1}+x_{2}+y_{1}=0$. By (1.13), via some direct computations, we have

[TABLE]

Noting that $mb$ is even, we obtain the folllowing identity for products of three theta functions:

[TABLE]

Taking $m=2$ in (4.36), we get

[TABLE]

Setting $x_{1}=y_{1}=y_{2}=0$ in (4.37), we get

[TABLE]

Taking $m=1$ in (4.36), we get

[TABLE]

Setting $x_{1}=y_{1}=y_{2}=0$ in (4.39), we get

[TABLE]

The above $b(\tau)$ and $b(y_{1},y_{2}|\tau)$ are defined by (4.2) and (4.6) respectively.

• •

Applications of Theorem 3.4.

Taking $n=3$ in (3.7) of Theorem 3.4, for $y_{1}+y_{2}+y_{3}=0$, by (3.8) we obtain

[TABLE]

We hence have

[TABLE]

Taking $m=2$ in (4.41), we get

[TABLE]

Setting $y_{1}=y_{2}=y_{3}=0$ in (4.42), we get

[TABLE]

The above $a(\tau)$ and $a(y_{1},y_{2}|\tau)$ are defined by (4.1) and (4.5) respectively.

Remark 4.1*.*

In [10], Zhu Cao obtained some identities for products of Ramanujan’s theta functions $f(a,b)$. No doubt many interesting identities of theta functions can be formulated from the Theorem 3.1–Theorem 3.4 and Theorem 3.11–Theorem 3.16, we here omit them.